Drunkard's Walk - Leonard Mlodinow [35]
The year was 1654. The question de Méré brought to Pascal was called the problem of points: Suppose you and another player are playing a game in which you both have equal chances and the first player to earn a certain number of points wins. The game is interrupted with one player in the lead. What is the fairest way to divide the pot? The solution, de Méré noted, should reflect each player’s chance of victory given the score that prevails when the game is interrupted. But how do you calculate that?
Pascal realized that whatever the answer, the methods needed to calculate it were yet unknown, and those methods, whatever they were, could have important implications in any type of competitive situation. And yet, as often happens in theoretical research, Pascal found himself unsure of, and even confused about, his plan of attack. He decided he needed a collaborator, or at least another mathematician with whom he could discuss his ideas. Marin Mersenne, the great communicator, had died a few years earlier, but Pascal was still wired into the Académie Mersenne network. And so in 1654 began one of the great correspondences in the history of mathematics, between Pascal and Pierre de Fermat.
In 1654, Fermat held a high position in the Tournelle, or criminal court, in Toulouse. When the court was in session, a finely robed Fermat might be found condemning errant functionaries to be burned at the stake. But when the court was not in session, he would turn his analytic skills to the gentler pursuit of mathematics. He may have been an amateur, but Pierre de Fermat is usually considered the greatest amateur mathematician of all times.
Fermat had not gained his high position through any particular ambition or accomplishment. He achieved it the old-fashioned way, by moving up steadily as his superiors dropped dead of the plague. In fact, when Pascal’s letter arrived, Fermat himself was recovering from a bout of the disease. He had even been reported dead, by his friend Bernard Medon. When Fermat didn’t die, an embarrassed but presumably happy Medon retracted his announcement, but there is no doubt that Fermat had been on the brink. As it turned out, though twenty-two years Pascal’s senior, Fermat would outlive his newfound correspondent by several years.
As we’ll see, the problem of points comes up in any area of life in which two entities compete. In their letters, Pascal and Fermat each developed his own approach and solved several versions of the problem. But it was Pascal’s method that proved simpler—even beautiful—and yet is general enough to be applied to many problems we encounter in our everyday experience. Because the problem of points first arose in a betting situation, I’ll illustrate the problem with an example from the world of sports. In 1996 the Atlanta Braves beat the New York Yankees in the first 2 games of the baseball World Series, in which the first team to win 4 games is crowned champion. The fact that the Braves won the first 2 games didn’t necessarily mean they were the superior team. Still, it could be taken as a sign that they were indeed better. Nevertheless, for our current purposes we will stick to the assumption that either team was equally likely to win each game and that the first 2 games just happened to go to the Braves.
Given that assumption, what would have been fair odds for a bet on the Yankees—that is, what was the chance of a Yankee comeback? To calculate it, we count all the ways in which the Yankees could have won and compare that to the number of ways in which